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In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut. A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience. The logarithm has a jump discontinuity of 2 π i when crossing the branch cut. The logarithm can be made ...
Any number log z defined by such criteria has the property that e log z = z. In this manner log function is a multi-valued function (often referred to as a "multifunction" in the context of complex analysis). A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π. These are the chosen ...
A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. The real part of log(z) is the natural logarithm of | z |. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.
The branch point for the principal branch is at z = − 1 / e , with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W −1 and W 1. In all branches W k with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.
This is commonly done by introducing a branch cut; in this case the "cut" might extend from the point z = 0 along the positive real axis to the point at infinity, so that the argument of the variable z in the cut plane is restricted to the range 0 ≤ arg(z) < 2π. We can now give a complete description of w = z 1/2.
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